20.

4 unique types of non-congruent triangles can be formed :

① Biggest equilateral △ ABC

② Small equilateral triangle
(CFE, FED, FAD and EDB are all congruent)

③ Obtuse scalene triangle
(FDB, FEB , EFA, EDA , DEC and DFC are all congruent)

④ Right-angled triangle
(CDA, CDB, BFC, BFA, AEC and AEB are all congruent)

21.

From 1001 to 1011, we notice the following 6 buddy pairs :


1001, 1002
1002, 1003
1003, 1004
1004, 1005
1009, 1010
1010, 1011


For 1011 to 1021, there are also 6 pairs.
For 1021 to 1031, also 6
For 1031 to 1041, also 6

But for 1041 to 1051, there are only 5.

(1049 + 1050 is the last possible here. We cannot do 1050 + 1051 as the tens place will carry over)

There is also no need to do 105_ ,106_ , 107_ , 108_ , 109_ EXCEPT :

1099 + 1100 = 2199
1100 + 1101 = 2201


Total number of pairs from 1001 to 1101

= 6 × 4 + 5 + 2
= 31

This forms 1 set.



We repeat the same for these sets :
1101 to 1201
1201 to 1301
1301 to 1401
1401 to 1501 (except 1500 + 1501, so this set only has 30 pairs)

There is no need to
try 1501 to 2000 as the hundreds place will carry over, EXCEPT :

1999 + 2000 = 3999

Total number of pairs

= 31 per set × 4 sets + 30 + 1 = 155

22.

x = 20182019 × 2020

= (20180000 + 2019) × 2020

= (2018 × 10000 + 2019) × 2020

= 2018 × 10000 × 2020 + 2019 × 2020



y = 20182020 × 2019

= (20180000 + 2020) × 2019

= (2018 × 10000 + 2020) × 2019

= 2018 × 10000 × 2019 + 2020 × 2019


Comparing x and y, we can see that their second products are the same but the first product is bigger for x.

So x > y



z = 20202019 × 2018

= (20200000 + 2019) × 2018

= (2020 × 10000 + 2019) × 2018

= 2020 × 10000 × 2018 + 2019 × 2018



Comparing y and z,

The second product is bigger for y
(bigger by 2 × 2018)

But, the first product is much bigger for z
(bigger by 1 × 10000 × 2018)

So z > y


Comparing x and z,

Their first product is the same but second product is bigger for x.

So x > z


∴ y < z < x (5)

23.


Primes :

2,3,5,7,11,13,17,19

There are 8 of them.

Composites :

4,6,8,9,10,12,14,15,16,18,20

There are 11 of them.



Number of ways to choose a prime number for the product

= 8

Number of ways to choose a composite for the product

= 11


Total number of possible products

= 8 × 11
= 88



Now notice a pattern :

For all products containing the prime number 2,

The sum = 2×4 + 2×6 + 2×8 + ... + 2×18 + 2×20

= 2 × (4 + 6 + 8 + ... + 18 + 20)

= 2 × 132


For a products containing prime number 3, the sum would be 3 × 132.

Likewise for products containing prime numbers 5,7,9,11...19


Sum of all products

= 2 × 132 + 3 × 132 + 5 × 132 + ... + 19 × 132

= (2+3+5+...+19) × 132

= 77 × 132

= 10164


This is just the product of the sum of primes and sum of composites.